# A02Discrete Parametrized Surfaces

## Developing a Theory of Discrete Surfaces with Constant Mean Curvature

In recent years, an exhaustive theory has been developed to understand and construct discrete minimal surfaces. We aim to produce something similar for the construction and classification of discrete surfaces with constant mean curvature (cmc). In particular, for discrete minimal surfaces the study of Koebe polyhedra that serve as Gauß maps has been fruitful - we are interested in analoga for discrete surfaces with constant mean curvature.

#### Scientific Details+

The aim of this project is to develop a theory of discrete surfaces of constant mean curvature (cmc surfaces for short) to an extent similar to what is known for discrete minimal surfaces. This includes understanding the structure and interconnections of various definitions of discrete cmc surfaces, their relationship with properties of their Gauß maps, construction methods, and hopefully existence and uniqueness results. Discretizing special classes of surfaces in R³ has been the starting point of many interesting developments in discrete differential geometry. Often the results have been isolated, but in some cases the the discretizations gave insight and ideas beyond its original scope and connections were made to different branches of discrete differential geometry. Discrete minimal surfaces are such a case, where among others integrable theory, combinatorics, surface discretization, and curvature theory via mesh parallelity all come together. We want to gain more insight into these interconnections by means of extending the theory - or rather complementing it - with a similar theory for cmc surfaces. In particular we plan to:

• Clarify the interrelations between several definitions of discrete cmc surfaces: There are several notions of discrete cmc surfaces all of which can be viewed as special cases of certain line congruence nets. Their interrelations are not well understood, though and the case of conical cmc surfaces has not been investigated at all.
• Investigate the structure of the Gauß maps of discrete cmc surfaces. The Gauß maps give rise to circle patterns on the sphere, which are interesting in their own right. Since the Gauß map of a smooth cmc surface is known to be harmonic, one can expect interesting analytic properties here. We hope to derive discrete versions of harmonicity for the various versions of discrete cmc surfaces.
• Find discrete analogues of the underlying partial differential equations and find analogues of Koebe's theorem for the above mentioned circle patterns: Cmc surfaces are governed by the sinh-Gordon equation and we expect to derive discrete integrable versions of the equation from the geometric definitions of discrete cmc surfaces. Likewise, we hope to find generalized versions of Koebe's theorem for the circle patterns that form the Gauß maps.
• Construction methods: It is unclear if one can expect something similar to the Weierstraß representation for (discrete) minimal surfaces. There, a variational principle allows solutions (and in the end minimal surfaces) to be constructed merely by prescribing the combinatorics. We will investigate whether one can apply similar mechanisms in case of discrete cmc surfaces. The Dorfmeister-Pedit-Wu method for cmc surfaces has been discretized for one flavour of discrete cmc surfaces, so we hope to be able to generalize that to the other variants as well.

Ideally the resulting theory should be comparable to the smooth theory and as rich and complete as the theory for discrete minimal surfaces, with (for example) its existence and uniqueness theorems.

However, the theory should not be developed just for its own sake. Not only should it be possible to extend the results to minimal surfaces in S³ via the so-called Lawson correspondence, but the intriguing interrelations between combinatorics, geometry and integrable systems - be found in the interrelations of Koebe polyhedra and discrete minimal surfaces - give hope for generalizations that not only give ways for constructing cmc surfaces but also produce insights beyond the primary agenda of this project.

One should note that this project will focus on integrable discretizations. This implies, that the we are concerned with discrete parametrized surfaces or "nets", not discretizations that come with arbitrary triangulated meshes.

#### Publications+

##### Papers
###### A discrete extrinsic and intrinsic Dirac operator

Authors: Hoffmann, Tim and Ye, Zi
Note: Preprint
Date: Feb 2018

###### Incircular nets and confocal conics

Authors: Akopyan, Arseniy and Bobenko, Alexander
Journal: Transactions of the American Mathematical Society, 370(4):2825--2854
Date: 2018
DOI: 10.1090/tran/7292

###### On a discretization of confocal quadrics. II. A geometric approach to general parametrizations

Authors: Bobenko, Alexander I and Schief, Wolfgang K and Suris, Yuri B and Techter, Jan
Note: Preprint
Date: 2017

###### On a discretization of confocal quadrics. I. An integrable systems approach

Authors: Bobenko, Alexander I. and Schief, Wolfgang K. and Suris, Yuri B. and Techter, Jan
Journal: Journal of Integrable Systems, 1(1):xyw005
Date: 2016
DOI: 10.1093/integr/xyw005

###### S-conical cmc surfaces. Towards a unified theory of discrete surfaces with constant mean curvature

Authors: Bobenko, A. I. and Hoffmann, T.
In Collection: Advances in Discrete Differential Geometry, Springer
Date: 2016

###### A 2x2 Lax representation, associated family, and Bäcklund transformation for circular K-nets

Authors: Hoffmann, T. and Sageman-Furnas, A. O.
Date: Oct 2015

###### S-conical minimal surfaces. Towards a unified theory of discrete minimal surfaces.

Authors: Bobenko, A. I. and Hoffmann, T. and König, B. and Sechelmann, S.
Note: Preprint
Date: 2015

###### A discrete parametrized surface theory in $\mathbb{R^3}$

Authors: Hoffmann, Tim and Sageman-Furnas, Andrew O. and Wardetzky, Max
Note: preprint
Date: Dec 2014

###### Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality

Authors: Bobenko, Alexander I. and Hertrich-Jeromin, Udo and Lukyanenko, Inna
Journal: Discrete and Computational Geometry, 52(4):612-629
Date: 2014
DOI: 10.1007/s00454-014-9622-5

###### Minimal surfaces from circle patterns: Geometry from combinatorics

Authors: Bobenko, A. and Hoffmann, T. and Springborn, B.
Journal: Ann. of Math., 164(1):231--264
Date: 2006

#### Prof. Dr. Tim Hoffmann   +

Projects: A02
University: TU München
E-Mail: hoffmant[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/TimHoffmann

#### Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, C01, B02, Z, CaP, II
University: TU Berlin, Institut für Mathematik, MA 881
Address: Strasse des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (30) 314 24655
E-Mail: bobenko[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~bobenko/

#### Benno König   +

Projects: A02
University: TU München

#### Alexander Preis   +

Projects: A02
University: TU Berlin
E-Mail: preis[at]math.tu-berlin.de

#### Jan Techter   +

Projects: A02
University: TU Berlin
E-Mail: techter[at]math.tu-berlin.de

#### Zi Ye   +

Projects: A02
University: TU München
E-Mail: ye[at]ma.tum.de